Υπομάδα
Υπομάς subgroup thumb|300px| [[Ομαδοθεωρία ---- Αλγεβρική Ομάδα Γενική Γραμμική Ομάδα Ορθογώνια Ομάδα Μοναδιακή Ομάδα ---- Μαθηματική Αναπαράσταση Μαθηματική Μήτρα ]] - Μία Ομάδα. Ετυμολογία Η ονομασία "ομάδα" σχετίζεται ετυμολογικά με την λέξη '"ομού". Εισαγωγή In group theory, a branch of mathematics, given a group ''G under a binary operation ∗, a subset H'' of ''G is called a subgroup of G'' if ''H also forms a group under the operation ∗. More precisely, H'' is a subgroup of ''G if the restriction of ∗ to H'' × ''H is a group operation on H''. This is usually denoted ''H ≤ G'', read as "''H is a subgroup of G''". The '''trivial subgroup' of any group is the subgroup {e''} consisting of just the identity element. A '''proper subgroup' of a group G'' is a subgroup ''H which is a proper subset of G'' (i.e.''H ≠ G''). This is usually represented notationally by ''H < G'', read as "''H is a proper subgroup of G''". Some authors also exclude the trivial group from being proper (i.e. {''e} ≠ H'' ≠ ''G).Hungerford (1974), p. 32Artin (2011), p. 43 If H'' is a subgroup of ''G, then G'' is sometimes called an '''overgroup' of H''. The same definitions apply more generally when ''G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G'' is sometimes denoted by the ordered pair (''G, ∗), usually to emphasize the operation ∗ when G'' carries multiple algebraic or other structures. This article will write ''ab for a'' ∗ ''b, as is usual. Basic properties of subgroups *A subset H'' of the group ''G is a subgroup of G'' if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever ''a and b'' are in ''H, then ab and a''−1 are also in ''H. These two conditions can be combined into one equivalent condition: whenever a'' and ''b are in H'', then ''ab−1 is also in H''.) In the case that ''H is finite, then H'' is a subgroup if and only if ''H is closed under products. (In this case, every element a'' of ''H generates a finite cyclic subgroup of H'', and the inverse of ''a is then a''−1 = ''an'' − 1, where ''n is the order of a''.) *The above condition can be stated in terms of a homomorphism; that is, ''H is a subgroup of a group G'' if and only if ''H is a subset of G'' and there is an inclusion homomorphism (i.e., i(''a) = a'' for every ''a) from H'' to ''G. *The identity of a subgroup is the identity of the group: if G'' is a group with identity ''eG'', and ''H is a subgroup of G'' with identity ''eH'', then ''eH'' = ''eG''. *The inverse of an element in a subgroup is the inverse of the element in the group: if ''H is a subgroup of a group G'', and ''a and b'' are elements of ''H such that ab = ba = e''H'', then ab = ba = e''G''. *The intersection of subgroups A'' and ''B is again a subgroup.Jacobson (2009), p. 41 The union of subgroups A'' and ''B is a subgroup if and only if either A'' or ''B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity. *If S'' is a subset of ''G, then there exists a minimum subgroup containing S'', which can be found by taking the intersection of all of subgroups containing ''S; it is denoted by <''S''> and is said to be the [[generating set of a group|subgroup generated by S'']]. An element of ''G is in <''S''> if and only if it is a finite product of elements of S'' and their inverses. *Every element ''a of a group G'' generates the cyclic subgroup <''a>. If <''a''> is isomorphic to Z'''/''nZ' for some positive integer n'', then ''n is the smallest positive integer for which a''n'' = e'', and ''n is called the order of a''. If <''a> is isomorphic to Z', then ''a is said to have infinite order. *The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) If e'' is the identity of ''G, then the trivial group {e''} is the minimum subgroup of ''G, while the maximum subgroup is the group G'' itself. under addition. The subgroup H contains only 0 and 4, and is isomorphic to \mathbb{Z}/2\mathbb{Z} . There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. The index : H is 4.]] Cosets and Lagrange's theorem Given a subgroup ''H and some a in G, we define the '''left coset aH = {ah : h'' in ''H}. Because a'' is invertible, the map φ : ''H → aH given by φ(h'') = ''ah is a bijection. Furthermore, every element of G'' is contained in precisely one left coset of ''H; the left cosets are the equivalence classes corresponding to the equivalence relation a''1 ~ ''a''2 if and only if ''a''1−1''a''2 is in ''H. The number of left cosets of H'' is called the index of ''H in G'' and is denoted by [''G : H'']. Lagrange's theorem states that for a finite group ''G and a subgroup H'', : [ G : H ] = { |G| \over |H| } where |''G| and |''H''| denote the orders of G'' and ''H, respectively. In particular, the order of every subgroup of G'' (and the order of every element of ''G) must be a divisor of |''G''|.See a didactic prove in this video. Right cosets are defined analogously: Ha = {ha : h'' in ''H}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G'' : ''H]. If aH = Ha for every a'' in ''G, then H'' is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if ''p is the lowest prime dividing the order of a finite group G, then any subgroup of index p'' (if such exists) is normal. Example: Subgroups of Z8 Let ''G be the cyclic group Z8 whose elements are : G=\left\{0,2,4,6,1,3,5,7\right\} and whose group operation is addition modulo eight. Its Cayley table is This group has two nontrivial subgroups: J''={0,4} and ''H={0,2,4,6}, where J'' is also a subgroup of ''H. The Cayley table for H'' is the top-left quadrant of the Cayley table for ''G. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic. Example: Subgroups of S4 (the symmetric group on 4 elements) Every group has as many small subgroups as neutral elements on the main diagonal: The trivial group and two-element groups Z2. Other examples *An ideal in a ring R is a subgroup of the additive group of R . *Let A be an abelian group; the elements of A that have finite period form a subgroup of A called the torsion subgroup of A . Υποσημειώσεις Εσωτερική Αρθρογραφία * Cartan subgroup * Fitting subgroup * Stable subgroup * Fixed-point subgroup * Subgroup test * Ομάδα * Ομαδοθεωρία * Ορθογώνια Ομάδα * Μοναδιακή Ομάδα * Ετεροτική Ομάδα * Ομαδιαία Αναπαράσταση * Μήτρα Βιβλιογραφία * . * . * . Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Κατηγορία:Μαθηματικές Ομάδες